# A small pond contains five catfish and four bluegill. If five fish are caught at random, what is the probability that all of them are catfish?

May 22, 2017

$\frac{15}{1890}$

#### Explanation:

$\text{there are a total of " 5+4=9" fish}$

$P \left(\text{catfish}\right) = \frac{5}{9}$

$\text{there are now a total of 8 fish, 4 are catfish}$

$P \left(\text{second catfish}\right) = \frac{4}{8} = \frac{1}{2}$

$\text{there are now 7 fish, 3 are catfish}$

$P \left(\text{third catfish}\right) = \frac{3}{7}$

$P \left(\text{fourth catfish}\right) = \frac{2}{6} = \frac{1}{3}$

$P \left(\text{fifth catfish}\right) = \frac{1}{5}$

$\text{to calculate the probability of catching 5 catfish}$
$\text{multiply the separate probabilities together }$

$\Rightarrow P \left(\text{5 catfish}\right) = \frac{5}{9} \times \frac{1}{2} \times \frac{3}{7} \times \frac{1}{3} \times \frac{1}{5} = \frac{15}{1890}$